📚 IGCSE Maths: Pythagoras, Bearings and Trigonometry | IGCSE 数学：勾股定理、方位角与三角学
In IGCSE Mathematics, Pythagoras' theorem, trigonometry and bearings appear across both core and extended papers. This revision guide explains the key formulas, how to choose the correct method, and the common exam pitfalls you must avoid.
在 IGCSE 数学中，勾股定理、三角学和方位角出现在核心与扩展试卷中。本复习指南讲解关键公式、如何选择正确方法以及考试中必须避免的常见错误。
1. Pythagoras' Theorem | 勾股定理
For any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b², where c is the longest side opposite the right angle.
对于任意直角三角形，斜边的平方等于另外两条边的平方和：c² = a² + b²，其中 c 是直角对面最长的边。
You can rearrange this to find a shorter side: a² = c² − b² or b² = c² − a².
你可以将其变形求出较短的边：a² = c² − b² 或 b² = c² − a²。
2. Labelling a Right-Angled Triangle | 标记直角三角形
Label the side opposite the angle θ as opposite, the side next to θ as adjacent, and the longest side as hypotenuse. Correct labelling is essential before using trigonometry.
将角 θ 对面的边标记为对边，与 θ 相邻的边标记为邻边，最长的边为斜边。在使用三角学之前，正确标记至关重要。
3. Trigonometric Ratios: SOH CAH TOA | 三角比：SOH CAH TOA
The three basic ratios are sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. Remember the phrase 'SOH CAH TOA'.
三个基本比值是 sin θ = 对边 ÷ 斜边、cos θ = 邻边 ÷ 斜边、tan θ = 对边 ÷ 邻边。记住短语 'SOH CAH TOA'。
- sin θ = O / H
- cos θ = A / H
- tan θ = O / A
Once you know two sides, choose the ratio that uses those two sides and the required angle.
一旦你知道两条边，就选择使用这两条边和所求角的比值。
4. Finding Missing Sides | 求未知边
tric ratio. For example, if the hypotenuse is 10 cm and θ = 30°, then the opposite side is 10 × sin 30° = 5 cm.
如果你有一条边和一个角，请使用合适的三角比。例如，如果斜边是 10 厘米且 θ = 30°，则对边为 10 × sin 30° = 5 厘米。
5. Finding Missing Angles | 求未知角
an angle, use the inverse trigonometric functions. If opposite = 4 and adjacent = 5, then θ = tan⁻¹(4 ÷ 5) ≈ 38.7°.
求角时，使用反三角函数。如果对边 = 4，邻边 = 5，则 θ = tan⁻¹(4 ÷ 5) ≈ 38.7°。
On your calculator, use sin⁻¹, cos⁻¹ or tan⁻¹ after dividing the two sides.
在计算器上，先将两条边相除，再使用 sin⁻¹、cos⁻¹ 或 tan⁻¹。
6. Angles of Elevation and Depression | 仰角与俯角
The angle of elevation is measured upwards from the horizontal. The angle of depression is measured downwards from the horizontal. Both often form right-angled triangles in exam diagrams.
仰角是从水平线向上测量的角；俯角是从水平线向下测量的角。在考试图中，两者通常构成直角三角形。
7. Bearings and Scale Drawings | 方位角与比例图
A bearing is a three-digit angle measured clockwise from North. For example, east is 090°, south is 180° and west is 270°.
方位角是从正北方向顺时针测量的三位数角度。例如，正东为 090°，正南为 180°，正西为 270°。
In scale drawings, use the scale factor to convert map lengths to real distances before solving triangles.
在比例图中，先使用比例尺将图上长度换算为实际距离，再求解三角形。
8. Sine Rule | 正弦定理
For any triangle ABC, the sine rule states a ÷ sin A = b ÷ sin B = c ÷ sin C or equivalently sin A ÷ a = sin B ÷ b = sin C ÷ c. Use it when you have two angles and one side, or two sides and a non-included angle.
对于任意三角形 ABC，正弦定理表示 a ÷ sin A = b ÷ sin B = c ÷ sin C，等价于 sin A ÷ a = sin B ÷ b = sin C ÷ c。当已知两角一边，或两边及一个非夹角时使用。
9. Cosine Rule | 余弦定理
The cosine rule is c² = a² + b² − 2ab cos C. Use it to find a side when you know two sides and the included angle, or to find an angle when you know all three sides.
余弦定理为 c² = a² + b² − 2ab cos C。当已知两边及其夹角时可求第三边；当已知三边时可求角度。
To find an angle, rearrange to cos C = (a² + b² − c²) ÷ 2ab.
求角时，变形为 cos C = (a² + b² − c²) ÷ 2ab。
10. Area of a Triangle Using Sine | 用正弦求三角形面积
For any triangle, Area = ½ ab sin C, where a and b are two sides and C is the included angle. This formula is especially useful for non-right-angled triangles.
对于任意三角形，面积 = ½ ab sin C，其中 a 和 b 是两条边，C 是夹角。该公式尤其适用于非直角三角形。
11. Which Rule to Use? | 如何选择规则
Use Pythagoras or SOH CAH TOA for right-angled triangles. Use the sine rule when a side is paired with its opposite angle. Use the cosine rule when you know two sides and the included angle or all three sides.
直角三角形使用勾股定理或 SOH CAH TOA。当边与其对角配对时使用正弦定理。当已知两边及夹角或三边时使用余弦定理。
12. Common Exam Mistakes and Tips | 常见考试错误与提示
Always write bearings with three digits, round only at the final step, check your calculator is in degree mode, and draw a clear diagram before choosing a formula.
方位角始终写成三位数；只在最后一步取整；检查计算器处于角度模式；在选择公式前先画出清晰的图。
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